Abstract

The solution of vector-valued Cahn–Hilliard systems is of interest in many applications. We discuss strategies for the handling of smooth and nonsmooth potentials as well as for different types of constant mobilities. Whereas the use of smooth potentials leads to a system of parabolic partial differential equalities, the nonsmooth ones result in variational inequalities. Concerning the latter, we propose a Moreau–Yosida regularization technique that incorporates the necessary bound constraints. As a result, the variational inequalities are replaced by nonsmooth equations. Due to the use of fully implicit time discretizations, which are the most accurate, we have to solve in every time step nonlinear smooth or nonsmooth equations. This is done by standard Newton methods in the smooth case, and by semismooth Newton methods in the nonsmooth case. At the heart of both methods lies the solution of large and sparse linear systems for which we propose the use of preconditioned Krylov subspace solvers using an effective Schur complement approximation. Numerical results illustrate the efficiency of our approach. In particular, we numerically show mesh and phase independence of the developed preconditioner in the smooth case. The results in the nonsmooth case are also satisfying and the preconditioned version always outperforms the unpreconditioned one. (Joint work with Martin Stoll.)

Abstract

We will introduce a whole range of problems related to random permutations whose motivation goes back to the Bose-Einstein condensation in quantum statistical mechanics. After reviewing standard probabilistic approaches we will introduce a Gibbs formalism for random bijections of the planar integer lattice. Under certain energy assumptions we show the existence of Gibbs measures and discuss possible characterisation of different phases and address the problem of finite and infinite cycles in bijections of the planar integer lattice.

Abstract

From the standpoint of basic fluid dynamics, katabatic winds are buoyantly driven boundary-layer-type flows along heated or cooled sloping surfaces in a stratified fluid. However, understanding their structure is of interest not only as a fundamental problem in itself, but also from a meteorological point of view, because of the broad range of areas and scales that they cover, influencing from local valleys micro-climate (e.g. over Salt Lake and Phoenix valleys) to synoptic scale motions (e.g. over ice sheets in coastal regions in Antarctica).

In katabatic flows turbulence is generated by shear and destroyed by negative buoyancy and viscosity. Because of this interplay between shear and buoyancy effects the strength of turbulence in the stable boundary layer that arise is much weaker, in comparison to the neutral and convective boundary layers, and this feature, together with the intrinsic complex dynamics of the system (e.g. occurrences of intermittency, Kelvin-Helmholtz instability, gravity waves, low-level jets and meandering motions) and the lack of any similarity theory, pose heavy burdens on numerical simulations.

The presentation will provide a brief overview of the state-of-the-art in numerical modeling of slope flows to then focus on recent numerical analyses, under idealized settings, which somehow resemble Prandtl's original model (1942) - an early milestone in the conceptual understanding of slope flows.

A modified set of filtered Boussinesq equations are solved on a regular domain relying on an operator-splitting technique to decouple the system. A mixed pseudo-spectral and finite difference approach is adopted in space and the fully explicit second-order accurate Adams-Bashforth scheme is used for time advancement. Closure of the equations is achieved through first order algebraic Smagorinsky models.

A statistical analysis of the initial oscillatory transient and on the properties of the steady state solution will be presented for a given subset of the parameter space, followed by an eduction of the coherent structures populating the flow. The behavior of Smagorinsky-type subgrid-scale models for such systems will also be discussed.

Abstract

The unit spheres in orthogonal representations of finite groups give examples of group actions on spheres. We investigate non-linear actions by studying chain complexes over the orbit category, and constructing finite G-CW complexes. This leads to new examples of homotopy representations with isotropy of rank one. This project is joint with Ergun Yalcin (Bilkent University, Ankara).

Abstract

The standard framework for studying representation theory of p-adic Lie groups is that of reductive groups over a p-adic field K. In this talk I will describe ongoing work with Clifton Cunningham and Takashi Suzuki where we instead work with limits of group schemes over the residue field of K. In particular, I will describe a sheaf-function dictionary for quasicharacters of tori over local fields, and early progress toward a definition for the affine Grassmannian for reductive groups over K.

Abstract

In order to construct the moduli space of canonical polarized manifolds, three different stability conditions have been introduced, namely, KSBA-stability, K-stabilty and asymptotic GIT stability. In this talk, we try to explore the relations among them. In particular, any canonical polarized manifold is stable with respect to all three conditions above, however the compactifications they give are different. As a consequence, we answer a longstanding question by showing that asymptotically GIT Chow semistable varieties do not form a proper family.

Abstract

We will first describe the conjecture of 'Quantum Unique Ergodicity' which is also known as 'Quantum Chaos'. This problem is in the intersection of Dynamical Systems, Harmonic Analysis, PDE, Differential Geometry and Mathematical Physics, and therefore attracts almost all branches of mathematics. Secondly, we will see why number theorists got immensely interested in this problem which is apparently coming from a different field of mathematics. We will also give a brief description of Lindenstrauss' ground-breaking work on this problem in a special case, for which he got Fields Medal in the ICM 2010.

Note for Attendees

Notice the special time and place. Last seminar of the academic year. Sushi will be served.

Abstract

We will introduce the basic notions of first order model theory, thestudy of first order theories and their structures. Many familiarclasses of mathematical objects are first order theories, such asgroups, fields, ZFC, some theories of arithmetic, and more. I willpresent some basic theorems and present an interesting result, Skolem'sParadox.

Note for Attendees

Notice the special time and place. Last seminar of the academic year. Sushi will be served.

Abstract

A linear algebraic group G defined over a field k is called special if every G-torsor over every field extension of k is trivial. In a modern language, it can be shown that the special groups are those of essential dimension zero. In 1958 Grothendieck classified special groups in the case where the base field k is algebraically closed. In this talk I will explain some recent progress towards the classification of special reductive groups over an arbitrary field. In particular, I will give the classification of special semisimple groups, special reductive groups of inner type and special quasisplit reductive groups over an arbitrary field k.

Note for Attendees

Note the unusual time of the seminar. This is in order to avoid a clash with the PIMS-CRM Fields lecture at 3pm.

Abstract

It is well known that Einstein's general theory of relativity provides a geometrical description of gravity in terms of space-time curvature. Einstein's theory poses some fascinating and difficult mathematical challenges that have stimulated a great deal of research in geometry and partial differential equations. Important questions include the well-posedness of the evolution problem, the definition of mass and angular momentum, the formation of black holes, the cosmic censorship hypothesis, the linear and non-linear stability of black holes and boundary value problems at conformal infinity arising in the analysis of the AdS/CFT correspondence. I will give a non-technical survey of some
significant advances and open problems pertaining to a number of these questions.

Abstract

I will describe a project to classify all smooth 4-dimensional manifolds triangulable with 6 or less 4-dimensional simplices. In the process we have found many simple triangulated 2-knot exteriors, forming a strong analogy with 3-manifold theory.

Abstract

The traditional framework in the atmospheric boundary layer for relating turbulent fluxes of momentum, heat, and scalar quantities to their mean gradients, called Monin-Obukhov similarity theory (MOST) after its originators, can be viewed as an extension of law of the wall scaling to account for the effects of thermal stratification. Although MOST is the standard framework for interpreting atmospheric measurements and modeling turbulent fluxes in weather and climate models, a number of fundamental issues in MOST still are not well understood. Because MOST arises from dimensional analysis, the connections between the curves that relate turbulent fluxes to mean gradients and fundamental properties of turbulence (e.g. the spectra, integral scales, and TKE budget) are not well understood. Furthermore, although experimental data often indicate deviations from MOST, the cause of these deviations (experimental error vs. physical processes) remains an open question.

In this presentation, a theoretical framework to connect MOST curves to fundamental properties of turbulence will be introduced. Experimental data will be used to demonstrate the effects of buoyancy on the integral length scales and their linkage to the behavior of MOST curves. Asymptotic solutions for MOST curves will also be derived for slightly unstable and free convective conditions.

In the second part of the talk, error propagation analysis and atmospheric data will be used to quantify the extent to which deviations from MOST are due to experimental errors vs. physical processes that are not represented by MOST. Deviations from MOST are found to have a strong diurnal trend, which suggest that processes related to the growth of the unstable atmospheric boundary layer remain unaccounted for in MOST.

The final part of the talk will focus on the how buoyancy and mean shear together influence the large-scale organization of the unstable atmospheric boundary layer. For slightly unstable conditions, convective updrafts organize into longitudinal rolls, aligned with the mean wind; for highly convective conditions, updrafts organize into cells, similar to Rayleigh-Benard convection. Using large eddy simulation, the transition from roll- to cellular- type convection will be examined. A transitional state between rolls and cells is observed and is characterized by oscillatory behavior in velocity statistics and convective organization. The physical processes responsible for this transition will be discussed.

Bio statement written by Scott: "I received my B.S. in Science Education from Martin Luther College in New Ulm, MN in 2008. In 2010, I received my M.S. in Meteorology from Penn State University. I will complete my Ph.D. (also in Meteorology) from Penn State University in May 2014, working with Prof. Marcelo Chamecki. My research interests include turbulence, the atmospheric boundary layer, and environmental fluid mechanics"

Abstract

I will discuss some recent developments in hyperbolic geometry and geometric group theory, namely Agol's proof of the virtual Haken conjecture and Wise's theory of special groups, together with their relationship with right-angled Artin groups and mapping class groups. I will then discuss a new result which shows that every hyperbolic 3-manifold admits a finite cover whose fundamental group embeds into a braid group, and into the group of diffeomorphisms of the circle. Finally, I will exhibit some higher dimensional closed hyperbolic manifold subgroups of braid groups and of the diffeomorphism group of the circle. The research in this talk represents work joint with Hyungryul Baik and Sang-hyun Kim.

Abstract

We discuss a result which shows that every right-angled Artin group quasi-isometrically embeds in a planar pure braid group. As a consequence, we obtain examples of quasi-isometrically embedded closed hyperbolic manifold subgroups of pure braid groups in all dimensions. We also give some applications to decision problems in braid group theory. This represents joint work with Sang-hyun Kim.

Abstract

Microlocal compactness forms (MCFs) are a new tool to study oscillations and concentrations in L^p-bounded sequences of functions. Decisively, MCFs retain information about the location, value distribution, and direction of oscillations and concentrations, thus extending both the theory of (generalized) Young measures and the theory of H-measures. Since in L^p-spaces oscillations and concentrations precisely discriminate between weak and strong compactness, MCFs allow to quantify the difference between these two notions of compactness. The definition involves a Fourier variable, whereby also differential constraints on the functions in the sequence can be investigated easily. Furthermore, pointwise restrictions are reflected in the MCF as well, paving the way for applications to Tartar's framework of compensated compactness; consequently, we establish a new weak-to-strong compactness theorem in a "geometric" way. Moreover, the hierarchy of oscillations with regard to slow and fast scales can be investigated as well since this information is also is reflected in the generated MCF.

Abstract

For any stationary mZ^d Gibbs measure that satisfies strong spatial mixing, we obtain sequences of upper and lower approximations that converge to its entropy. In the case d=2, these approximations are efficient in the sense that they are accurate to within epsilon and can be computed in time polynomial in 1/epsilon. The method is extended to approximate pressure of Gibbs interactions. Joint work with Ronnie Pavlov.